Posterior Summary and Visualization Utilities

This vignette demonstrates the summary and plotting utilities available for stochtree models.

Setup

Load necessary packages

library(stochtree)

import numpy as np
import matplotlib.pyplot as plt
from stochtree import BARTModel, BCFModel, plot_parameter_trace

Set a seed for reproducibility

random_seed = 1234
set.seed(random_seed)

random_seed = 1234
rng = np.random.default_rng(random_seed)

Supervised Learning

We begin with the supervised learning use case served by the bart() function.

Below we simulate a simple regression dataset.

n <- 1000
p_x <- 10
p_w <- 1
X <- matrix(runif(n * p_x), ncol = p_x)
W <- matrix(runif(n * p_w), ncol = p_w)
f_XW <- (((0 <= X[, 10]) & (0.25 > X[, 10])) *
  (-7.5 * W[, 1]) +
  ((0.25 <= X[, 10]) & (0.5 > X[, 10])) * (-2.5 * W[, 1]) +
  ((0.5 <= X[, 10]) & (0.75 > X[, 10])) * (2.5 * W[, 1]) +
  ((0.75 <= X[, 10]) & (1 > X[, 10])) * (7.5 * W[, 1]))
noise_sd <- 1
y <- f_XW + rnorm(n, 0, 1) * noise_sd

n = 1000
p_x = 10
p_w = 1
X = rng.uniform(size=(n, p_x))
W = rng.uniform(size=(n, p_w))
# R uses X[,10] (1-indexed) = Python X[:,9]
f_XW = (
    ((X[:, 9] >= 0)    & (X[:, 9] < 0.25)) * (-7.5 * W[:, 0]) +
    ((X[:, 9] >= 0.25) & (X[:, 9] < 0.5))  * (-2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.5)  & (X[:, 9] < 0.75)) * ( 2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.75) & (X[:, 9] < 1.0))  * ( 7.5 * W[:, 0])
)
noise_sd = 1.0
y = f_XW + rng.standard_normal(n) * noise_sd

Now we fit a simple BART model to the data.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3
)
bart_model <- stochtree::bart(
  X_train = X,
  y_train = y,
  leaf_basis_train = W,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bart_model = BARTModel()
bart_model.sample(
    X_train=X,
    y_train=y,
    leaf_basis_train=W,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3
    },
)

We obtain a high level summary of the BART model by running print().

print(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bart_model)

BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function.

summary(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.894, standard deviation = 0.049, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.8019795 0.8328433 0.8599583 0.8936937 0.9257733 0.9562986 0.9943913 
Summary of leaf scale posterior: 
3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
       2.5%         10%         25%         50%         75%         90% 
0.005034958 0.005617630 0.006155038 0.006797218 0.007540116 0.008477232 
      97.5% 
0.010059888 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = -0.063, standard deviation = 3.285, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-6.8051628 -4.9258391 -1.8289427 -0.1191959  2.0089641  4.2799876  6.5376163

print(bart_model.summary())

BART Model Summary:
-------------------
BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.935, standard deviation = 0.048, quantiles:
    2.5%: 0.846
   10.0%: 0.876
   25.0%: 0.903
   50.0%: 0.934
   75.0%: 0.966
   90.0%: 0.997
   97.5%: 1.037
Summary of leaf scale posterior: 3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
    2.5%: 0.005
   10.0%: 0.006
   25.0%: 0.006
   50.0%: 0.007
   75.0%: 0.008
   90.0%: 0.008
   97.5%: 0.009
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 0.106, standard deviation = 3.287, quantiles:
    2.5%: -6.692
   10.0%: -4.343
   25.0%: -1.948
   50.0%: 0.169
   75.0%: 2.095
   90.0%: 4.415
   97.5%: 6.584

None

We can use the plot() function to produce a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

plot(bart_model)

ax = plot_parameter_trace(bart_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function to pull the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), leaf scale \(\sigma^2_{\ell}\), in-sample mean function predictions y_hat_train) and then plot any subset or transformation of these values.

y_hat_train_samples <- extractParameter(bart_model, "y_hat_train")
obs_index <- 1
plot(
  y_hat_train_samples[obs_index, ],
  type = "l",
  main = paste0("In-Sample Predictions Traceplot, Observation ", obs_index),
  xlab = "Index",
  ylab = "Parameter Values"
)

y_hat_train_samples = bart_model.extract_parameter("y_hat_train")
obs_index = 0
fig, ax = plt.subplots()
ax.plot(y_hat_train_samples[obs_index, :])
ax.set_title(f"In-Sample Predictions Traceplot, Observation {obs_index}")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
plt.show()

Causal Inference

We now run the same demo for the causal inference use case served by the bcf() function in R and the BCFModel Python class.

Below we simulate a simple dataset for a causal inference problem with binary treatment and continuous outcome.

# Generate covariates and treatment
n <- 1000
p_X = 5
X = matrix(runif(n * p_X), ncol = p_X)
pi_X = 0.25 + 0.5 * X[, 1]
Z = rbinom(n, 1, pi_X)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[, 3]
tau_X = X[, 2] * 2 - 1

# Generate outcome
epsilon = rnorm(n, 0, 1)
y = mu_X + tau_X * Z + epsilon

# Generate covariates and treatment
n = 1000
p_X = 5
X = rng.uniform(size=(n, p_X))
pi_X = 0.25 + 0.5 * X[:, 0]
Z = rng.binomial(1, pi_X, n).astype(float)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[:, 2]
tau_X = X[:, 1] * 2 - 1

# Generate outcome
epsilon = rng.standard_normal(n)
y = mu_X + tau_X * Z + epsilon

Now we fit a simple BCF model to the data

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3, 
  adaptive_coding = TRUE
)
bcf_model <- stochtree::bcf(
  X_train = X,
  y_train = y,
  Z_train = Z,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bcf_model = BCFModel()
bcf_model.sample(
    X_train=X,
    Z_train=Z,
    y_train=y,
    propensity_train=pi_X,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3, 
      "adaptive_coding": True
    },
)

We obtain a high level summary of the BCF model by running print().

print(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bcf_model)

BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function / method.

summary(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.950, standard deviation = 0.045, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.8671638 0.8955517 0.9194418 0.9482912 0.9778809 1.0082480 1.0432562 
Summary of prognostic forest leaf scale posterior: 
3000 samples, mean = 0.001, standard deviation = 0.000, quantiles:
        2.5%          10%          25%          50%          75%          90% 
0.0009155494 0.0010444923 0.0011923705 0.0013832633 0.0016050074 0.0018443221 
       97.5% 
0.0022083584 
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.474, mean (treated) = 0.999, standard deviation (control) = 0.360, standard deviation (treated) = 0.322
quantiles (control):
       2.5%         10%         25%         50%         75%         90% 
-1.22538042 -0.94537274 -0.71587579 -0.44515320 -0.20584871 -0.02614317 
      97.5% 
 0.15561358 
quantiles (treated):
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.3822691 0.5906006 0.7787172 0.9926902 1.2086486 1.4161943 1.6394560 
Summary of treatment effect intercept (tau_0) posterior: 
3000 samples, mean = 0.086, standard deviation = 0.788, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-1.9032719 -1.0172604 -0.3145133  0.1186374  0.6614863  1.0476190  1.4041918 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.487, standard deviation = 0.940, quantiles:
    2.5%      10%      25%      50%      75%      90%    97.5% 
1.712540 2.287648 2.810896 3.480523 4.136114 4.709252 5.432332 
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = 0.077, standard deviation = 0.519, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-0.7188719 -0.5463845 -0.3848017 -0.0175150  0.5787601  0.7923522  0.9163277

print(bcf_model.summary())

BCF Model Summary:
------------------
BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.875, standard deviation = 0.043, quantiles:
    2.5%: 0.795
   10.0%: 0.820
   25.0%: 0.845
   50.0%: 0.874
   75.0%: 0.904
   90.0%: 0.929
   97.5%: 0.959
Summary of prognostic forest leaf scale posterior: 3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
    2.5%: 0.001
   10.0%: 0.001
   25.0%: 0.001
   50.0%: 0.002
   75.0%: 0.002
   90.0%: 0.002
   97.5%: 0.002
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.489, mean (treated) = 1.161, standard deviation (control) = 0.307, standard deviation (treated) = 0.316
quantiles (control):
    2.5%: -1.135
   10.0%: -0.892
   25.0%: -0.693
   50.0%: -0.469
   75.0%: -0.276
   90.0%: -0.096
   97.5%: 0.059

quantiles (treated):
    2.5%: 0.560
   10.0%: 0.754
   25.0%: 0.940
   50.0%: 1.162
   75.0%: 1.361
   90.0%: 1.567
   97.5%: 1.802
Summary of treatment effect intercept (tau_0) posterior: 3000 samples, mean = 0.611, standard deviation = 0.470, quantiles:
    2.5%: -0.286
   10.0%: -0.043
   25.0%: 0.265
   50.0%: 0.660
   75.0%: 0.937
   90.0%: 1.211
   97.5%: 1.497
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.482, standard deviation = 0.960, quantiles:
    2.5%: 1.838
   10.0%: 2.212
   25.0%: 2.772
   50.0%: 3.474
   75.0%: 4.158
   90.0%: 4.757
   97.5%: 5.318
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = -0.028, standard deviation = 0.625, quantiles:
    2.5%: -1.075
   10.0%: -0.919
   25.0%: -0.573
   50.0%: 0.054
   75.0%: 0.543
   90.0%: 0.750
   97.5%: 0.927

None

In R, we have a plot() that produces a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

In Python, we provide a plot_parameter_trace() function for requesting a traceplot of a specific model parameter.

plot(bcf_model)

ax = plot_parameter_trace(bcf_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function in R or the extract_parameter() method in Python to query the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), prognostic forest leaf scale \(\sigma^2_{\mu}\), CATE forest leaf scale \(\sigma^2_{\tau}\), adaptive coding parameters \(b_0\) and \(b_1\) for binary treatment, in-sample mean function predictions y_hat_train, in-sample CATE function predictions tau_hat_train) and then plot any subset or transformation of these values.

adaptive_coding_samples <- extractParameter(bcf_model, "adaptive_coding")
plot(
  adaptive_coding_samples[1, ],
  type = "l",
  main = "Adaptive Coding Parameter Traceplot",
  xlab = "Index",
  ylab = "Parameter Values",
  ylim = range(adaptive_coding_samples),
  col = "blue"
)
lines(adaptive_coding_samples[2, ], col = "orange")
legend(
  "topright",
  legend = c("Control", "Treated"),
  lty = 1,
  col = c("blue", "orange")
)

adaptive_coding_samples = bcf_model.extract_parameter("adaptive_coding")
fig, ax = plt.subplots()
ax.plot(adaptive_coding_samples[0, :], color="blue", label="Control")
ax.plot(adaptive_coding_samples[1, :], color="orange", label="Treated")
ax.set_title("Adaptive Coding Parameter Traceplot")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
ax.legend(loc="upper right")
plt.show()